This invention relates to methods and apparatus for divergent beam tomography, and more particularly to methods and apparatus for divergent beam tomography that reduce computational complexity without visually perceptible degradation or significant numerical inaccuracy.
Tomographic reconstruction is a well-known technique underlying nearly all of the diagnostic imaging modalities, including x-ray computed tomography (CT), positron emission tomography (PET), single photon emission tomography (SPECT), and certain acquisition methods for magnetic resonance imaging (MRI). It is also widely used in manufacturing for nondestructive evaluation (NDE), and more recently for airport baggage inspection. The dominant data acquisition mode in current CT scanners is the fan-beam geometry, often combined with a helical or spiral scan.
In the interest of faster image acquisition, CT scanner technology is moving to cone-beam acquisition using area detectors. Already many of the CT scanners produced have a multirow detector (typically 16 rows), resulting in (low-angle) cone-beam acquisition. Area detectors for wide-angle cone-beam acquisition will be introduced in medical CT scanners over the next few years. Currently there is an unmet need for high-speed, 3D tomographic imaging for detection of concealed weapons and explosive devices. In other modalities, such as PET or SPECT, cone-beam acquisition is already the dominant format.
The reconstruction problem in fan-beam and in cone-beam CT is to recover a 2D or 3D image of an object from a set of its line-integral projections along lines diverging as a fan or a cone, respectively, from a source traveling on a specified acquisition trajectory or orbit. The fan-beam and cone-beam geometries are therefore instances of the divergent-beam geometry.
The method of choice for tomographic reconstruction is filtered backprojection (FBP) or convolution backprojection (CBP), both of which use a weighted backprojection step. This step is the computational bottleneck in the technique, with computational requirements scaling as N3 for an Nxc3x97N-pixel image in 2D, and at least as N4 for an Nxc3x97Nxc3x97N-voxel image in 3D. Thus, doubling the image resolution from N to 2N results in roughly an 8-fold (or 16-fold, in 3D) increase in computation. While computers have become much faster, with the advent of new technologies capable of collecting ever larger quantities of data in real time (e.g., cardiac imaging with multi-row detectors, interventional imaging), and the move to 3D cone-beam geometry, there is still a growing need for fast reconstruction techniques. Fast reconstruction can speed up the image formation process, reduce the cost of a special-purpose image reconstruction computer (see below), or both.
The dual operation of backprojection is reprojection, which is the process of computing the projections of an electronically stored image. This process, too, plays a fundamental role in tomographic reconstruction. A combination of backprojection and reprojection can also be used to construct fast reconstruction algorithms for the long object problem in the helical cone-beam geometry, which is key to practical 3D imaging of human subjects. Furthermore, in various applications it is advantageous or even necessary to use iterative reconstruction algorithms, in which both backprojection and reprojection steps are performed several times for the reconstruction of a single image. Speeding up the backprojection and reprojection steps will determine the economic feasibility of such iterative methods.
In discussing fast methods, it is important to distinguish between the two main formats for tomographic data: (i) parallel-beam; and (ii) divergent-beam and its special cases. While the parallel-beam format is the most convenient for both theoretical and computational manipulations, divergent-beam is the format most commonly found in commercial medical and industrial CT scanners. Although originally a 2D method, fan-beam reconstruction is the key component in state-of-the-art methods for helical and multi-slice helical 3D (volumetric) reconstruction. New and future designs are expected to migrate to the cone-beam geometry. Thus, whether in 2D or 3D, the divergent-beam geometry is the dominant imaging format, and will most likely remain so for the foreseeable future.
In the limit of large source distance from the object, the divergent-beam geometry reduces to the parallel-beam geometry. Therefore, all data processing methods for the divergent-beam geometry (including the methods of this invention) are also applicable to the parallel-beam geometry. However, the converse is not true: for the small or moderate source distances found in practice, the two geometries are sufficiently different, that parallel-beam processing methods yield inferior or unsatisfactory results when applied directly to divergent-beam data. Thus, divergent-beam geometries require processing different from parallel-beam geometries.
The present invention addresses divergent-beam tomography. Some reconstruction methods rely on a process called rebinning, to rearrange (or interpolate) divergent-beam data into parallel-beam data, which is then processed by a parallel-beam algorithm. Other methods for 3D divergent-beam geometries rely on transforming the cone-beam data to 3D Radon transform data (or a derivative thereof). Instead, native divergent-beam methods (of which the present invention is a fast version) by definition process the divergent-beam data directly, without prior rebinning to parallel-beam data or transforming the data to the 3D Radon transform domain. The reconstruction process in native divergent-beam methods consists of preprocessing the projections (e.g., weighting and filtering), followed by a weighted divergent-beam backprojection operation and perhaps a divergent-beam reprojection operation, or a series of such operations. Such methods are therefore called divergent-beam filtered backprojection (DB-FBP) algorithms. Their computation is usually dominated by the backprojection and reprojection operations, just as for the parallel-beam case.
Divergent-beam reprojection can be performed by first performing a parallel-beam reprojection, and then rebinning the results, but here too, it is advantageous to use a native divergent-beam reprojection algorithm, which does not use rebinning.
The drawbacks of rebinning include possible artifacts and added computation, which limit the speedup. Rebinning also requires acquisition and manipulation of a large part of the data before processing can start, once again limiting speed. Methods that rely on a transformation to the 3D Radon domain have similar drawbacks. Thus, there is a need for methods and apparatus for divergent-beam tomography that overcome these drawbacks, that are highly flexible, and that provide large performance gains compared to methods that use conventional divergent-beam backprojection.
Special-purpose hardware has been the traditional avenue to speeding up the backprojection process. Special types of computers, using custom chips, or multiple processors, or combinations thereof, are designed to perform the necessary computations.
Drawbacks of this approach include the cost of multiple processors or custom hardware, and the fact that with the high rate of increase in the performance of general purpose computers, the special purpose architectures required quickly become obsolete. Thus, there is a need for fast processes for the divergent-beam geometry that do not require special-purpose hardware, and that are easily implemented on standard serial or parallel architectures, to make them more cost effective.
Accordingly, one object of this invention is to provide new and improved methods and apparatus for divergent-beam tomography.
Another object is to provide new and improved methods and apparatus for divergent-beam tomography that reduce computational complexity without visually perceptible degradation or significant numerical inaccuracy.
In keeping with one aspect of this invention, a method is proposed for generating an electronic image from a preprocessed divergent-beam sinogram which is amenable to backprojection, the sinogram being a collection of divergent beam projections. The method can include the following steps: subdividing the sinogram into a plurality of sub-sinograms; performing a weighted backprojection of the sub-sinograms in the global coordinate system fixed to the object to produce a plurality of corresponding sub-images at proper locations in the global coordinate system; and aggregating the sub-images to create the electronic image. The subdivision of the sinogram into sub-sinograms can be performed in a recursive manner until each sub-sinogram represents a sub-image of a desired size, where the subdividing steps include a desired number of exact subdivisions and a desired number of approximate subdivisions.
In keeping with another aspect of this invention, a method is proposed for reprojecting an electronic image, that is, generating a divergent-beam sinogram from the image. The method can include the following steps: dividing the image into a plurality of sub-images; computing sub-sinograms of each sub-image in the global coordinate system; and aggregating the sub-sinograms to create the sinogram. The subdivision of the image can be performed in a recursive manner until each sub-image has a desired size, and the computation of sub-sinograms can be approximate in a desired number of levels in the recursion, and exact in the remaining levels of the recursion.
These methods offer speedups similar to the FFT when compared to conventional backprojection and reprojection algorithms. Substantial computational savings are obtained without perceptible degradation or significant loss of numerical accuracy.